Are the following two assertions always true, always false or meaningless?
$\exists i \in \emptyset$
$\forall i \in \emptyset$
Because it seems that one encounters expressions of this kind fairly similar in mathematics, especially if we are dealing with degenerate cases of definitions. Let me give an example (in graph theory) of such a case. There, one can formalize the idea the two vertices are connected in the following way: Let $G=(V,E)$ be a graph and let $v,w\in V $. We define $v$ and $w$ to be "connected" if: $\exists n \in \mathbb{N}, \ \exists \alpha: \left\{ 1,\ldots,n \right\} \rightarrow V, \ \alpha_0=v \ \& \ \alpha_n=w \ \ \forall i\in \mathbb{N}, 0\leqslant i \ \& \ i<n:\ \left\{ \alpha_i, \alpha_{i+1} \right\} \in E$
Now, if we would ask, if every node is connected to itself (a fact which intuitively we would want to be true), we would have to exhibit an $n$ and a sequence $\alpha$, such that [bla bla...]. The obvious choice for $n$ is 0. But for this choice of $n$, no mattter which sequence $\alpha$ we would consider, the set of the $i$'s would be empty, because the set of the $i$'s is actually $\left\{ i\in \mathbb{N} | 0\leqslant j \ \& \ j<n \right\}$. But for $n=0$ this set is the empty set. Thus, because we quantify $\forall i \in \left\{ i\in \mathbb{N} | 0\leqslant j \ \& \ j<0 \right\}=\emptyset$, should the statement be true/false by definition?
EDIT: Sorry, for my unclear formulation and to everybody who on my fault interpreted it wrongly. The way Carl Mummert or Listing interpreted it, was what I meant.